Arnold diffusion, as first exposed in [1]} has become a major subject of study for nearly integrable Hamiltonian systems. This mechanism epitomises how an integrable (“stable”) system can become unstable, with actions varying slowly (as permitted by Nekhoroshev's theorem) but substantially, as opposed to what would happen for an integrable perturbation [2]}, [3]}. A key idea has been to focus on a normally hyperbolic invariant cylinder. The cylinder is indeed a more robust object than individual hyperbolic invariant tori that may lie inside the cylinder and which Arnold initially used. Another key addition to Arnold's mechanism is the random iteration of the dynamics carried by the cylinder and the “outer” dynamics obtained at the limit by following unstable and stable leaves of the cylinder (see figure REF ). Both these ideas appear in Moeckel's work [4]}.
These ideas were formalised by Delshams, de la Llave and Seara with the invention of the scattering map which encodes the outer dynamics. This allowed them to solve the large gap problem, when two (primary) invariant tori are too far away for their unstable and stable manifolds to meet [5]}. In these last decades, this approach has been successfully implemented by a number of authors, using either a geometric description [5]}, [7]}, [8]}, [9]}, [10]}, [11]} or a variational one [12]}, [13]}.
